A.T. Fomenko, Double cover of the Klein bottle by torus
Pasha Zusmanovich

6/7TOPO Topology, winter semester 2025/2026

Monday 10:50-12:25   G302   ("Lectures" only)

Exam terms:
Thursday December 18  09:00   G302
FridayDecember 19  09:00   G503
MondayDecember 22  09:00   G503
Rules for taking exams


LITERATURE
Main: Additional: All the books are available in electronic form in multiple places.          = available in the university library

A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours, unless specified otherwise; tutorials are running separately)

Class 1 September 22, 2025
Organizational details. The subject of topology. Definition of a topological space. Examples of topological spaces: trivial and discrete toplogy, topologies on 2- and 3-element sets, the standard topology on \(\mathbb R\). Number of topologies on a finite set.
([V], pp. xi-xii, 2, 11-12; Munkres, p.76).

Class 2 September 29, 2025
Base of a topology, examples. Any base of the standard topology on \(\mathbb R\) can be decreased. Criterion for a family of open sets to be a base ([V], p.16, 3.A).

Class 3 October 6, 2025
Further criteria for a family of subsets to be a base ([V], 3.B, 3.C). Finer and coarser topologies. Criterion for a topology to be coarser (Munkres, Lemma 13.3).
([V], pp.16-17; Munkres, p.81).

Class 4 October 13, 2025
The standard topology on \(\mathbb R^2\). Closed sets, definition of topology in terms of closed sets. Clopen sets. A parody of an episode from the movie "The Bunker" featuring clopen sets.
([V], pp.13-14,16-17).

Class 5 October 20, 2025
Descritption of topologies having exactly one base. Metric spaces. Euclidean metric, Manhattan metric, 0-1 metric, finite metric spaces. Restriction of a metric to a subspace. Balls and spheres. Metric topology.
([V], pp.16,18-20,22; Munkres, pp.119-120).

Class 6 October 27, 2025
Proof that the metric topology is well-defined (open balls form a base of the topology). Examples of metric topologies. Metrizability of topological spaces, examples of metrizable and non-metrizable spaces. Hausdorff topological space. Metric topologies are Hausdorff. The arrow.
([V], pp. 11-12,22,97; Munkres, pp.119-123).

Class 7 November 10, 2025 (3 hours)
Topological and metric equivalences of metrics. Metric equivalence implies topological equivalence. If \(\rho\) is a metric, then \(\frac{\rho}{1+\rho}\) is topologically equivalent, but not necessary metrically equivalent, metric. pdf TeX
Subspace topology, examples. Interior, closure, boundary.
([V], pp.22-23,27,29-31).

Class 8 November 24, 2025 (3 hours)
The boundary of a set coincides with the set of boundary points (remake). Definition of continuous and open (i.e., an image of an open set is open) maps. \(x \mapsto |x|\) is continuous but not open. The continuity of a map \(f: X \to Y\) is equivalent to each of the following conditions: \[ f^{-1}(Cl(B)) \supseteq Cl(f^{-1}(B)) \text{ for any } B \subseteq Y \\ f^{-1}(Int(B)) \subseteq Int(f^{-1}(B)) \text{ for any } B \subseteq Y \\ f(Cl(A)) \subseteq Cl(f(A)) \text{ for any } A \subseteq X \\ f(Int(A)) \supseteq Int(f(A)) \text{ for any } A \subseteq X . \] Continuity at a point. Equivalence with the \(\epsilon-\delta\) definiton of continuity from analysis in the case of metric topologies. Homeomorphisms (definition, elementary properties).
([V], pp.30,59-61,67-69; Munkres, pp.102-105).


A brief synopsis, videos, and homeworks from this course at the previous semesters

Created: Fri Oct 2 2020
Last modified: Mon Nov 24 2025 18:26:41 CET